A combinatorial application of Alexander duality

百花草

The M¨obius number of a finite partially ordered set equals (up to sign) the
difference between the number of even and odd edge covers of its incomparability
graph. One way to deduce this formula uses Stanley’s combinatorial Alexander
duality theorem for Eulerian posets and Rota’s cross-cut theorem for lattices.
Thereby, the formula may be viewed as a consequence of two theorems from algebraic
topology: Alexander Duality and the Nerve Theorem. We use these theorems
to obtain a refinement that relates the homology of a poset’s order complex to the
cohomology of its incomparability complex, whose simplices are sets of edges of
its incomparability graph that do not cover.

zdr2005

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